Abstract

We give a quantum algorithm that finds collisions in arbitrary r-to-one functions after only O(3√N/r) expected evaluations of the function, where N is the cardinality of the domain. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Further, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way.